#8993992

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,14)(13,16)(17,18,22,24)(19,27,28,23)(20,29,30,25)(21,26,31,32)(33,34)(35,40)(36,39)(37,41)(38,42)(43,45)(44,47)(46,48);(1,3,6,9)(2,7,5,11)(4,10,12,13)(8,15,14,16)(17,19,22,28)(18,23,24,27)(20,21,30,31)(25,26,29,32)(33,35)(34,36)(37,43)(38,39)(40,42)(41,44)(45,48)(46,47);(2,5)(3,9)(4,12)(15,16)(18,24)(20,30)(21,31)(23,27)(33,36,38,40)(34,39,42,35)(37,45,46,44)(41,43,48,47);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,13)(11,16)(17,20,22,30)(18,25,24,29)(19,21,28,31)(23,26,27,32)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,46)(47,48);(1,3)(2,7)(4,13)(5,11)(6,9)(8,16)(10,12)(14,15)(17,21)(18,26)(19,20)(22,31)(23,25)(24,32)(27,29)(28,30)(33,35)(34,40)(36,42)(37,43)(38,39)(41,45)(44,48)(46,47);(1,5,6,2)(3,11,9,7)(4,14,12,8)(10,16,13,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,30)(31,32)(33,37,38,46)(34,41,42,48)(35,43,39,47)(36,44,40,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8901188);